Cyclic orbifolds of lattice vertex operator algebras having group like fusions

Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_{L}^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^*/L$ (or equivalently $(1-g)L^*<L$). We also determine their fusion rings and the corresponding quadratic space structures when $g$ is fixed point free on $L$. By applying our method to some coinvariant sublattices of the Leech lattice $\Lambda$, we prove a conjecture proposed by G. H\"ohn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge $24$ using the the orbifold vertex operator algebra $V_{\Lambda_g}^{\hat{g}}$.Comment: The main theorem was proved in a slightly more general setting and the title of the article has been change

A characterization of the moonshine vertex operator algebra by means of Virasoro frames

In this article, we show that a framed vertex operator algebra V satisfying the conditions: (1) V is holomorphic (i.e., V is the only irreducible V-module); (2) V is of rank 24; and (3) V_1=0; is isomorphic to the moonshine vertex operator algebra constructed by Frenkel-Lepowsky-Meurman.Comment: 10 pages, no figur

On the structure of framed vertex operator algebras and their pointwise frame stabilizers

In this paper, we study the structure of a general framed vertex operator algebra. We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V_C. This result would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous Moonshine VOA are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of the moonshine VOA is isomorphic to the moonshine VOA itself.Comment: Version 3: 59 pages. Corrected version. 54 pages on my LaTeX system version 2: We add Theorem 5.16 in which we give a necessary and sufficient condtion for a code to be a structure code of a holomorphic framed VOA. "hyperref" style is also introduce

Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24

In 1993, Schellekens obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras using the theory of framed vertex operator algebras and to determine the Lie algebra structures of their weight one subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes \RM(1,4)^3 and \RM(1,4)\oplus \EuD(d_{16}^+) of length 48. These vertex operator algebras correspond to the holomorphic simple current extensions of the lattice type vertex operator algebras $(V_{\sqrt{2}E_8}^+)^{\otimes 3}$ and $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. We determine such extensions using a quadratic space structure on the set of all irreducible modules $R(W)$ of $W$ when $W= (V_{\sqrt{2}E_8}^+)^{\otimes 3}$ or $V_{\sqrt{2}E_8}^+\otimes V_{\sqrt{2}D_{16}^+}^+$. As our main results, we construct seven new holomorphic vertex operator algebras of central charge 24 in Schellekens' list and obtain a complete list of all Lie algebra structures associated to the weight one subspaces of holomorphic framed vertex operator algebras of central charge 24.Comment: 46 page

On 3-transposition groups generated by σ\sigma-involutions associated to c=4/5 Virasoro vectors

In this paper, we show that $\sigma$-involutions associated to extendable c=4/5 Virasoro vectors generate a 3-transposition group in the automorphism group of a vertex operator algebra (VOA). Several explicit examples related to lattice VOA are also discussed in details. In particular, we show that the automorphism group of the VOA $V_{K_{12}}^{\hat{\nu}}$ associated to the Coxeter Todd lattice $K_{12}$ contains a subgroup isomorphic to ${}^+\Omega^{-}(8,3)$.Comment: 38 page

The Conway-Miyamoto correspondences for the Fischer 3-transposition groups

In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras of the moonshine vertex operator algebra.Comment: 3 figure files (fig1.tex, fig2.tex, fig3.tex) include

Level-Rank Duality for Vertex Operator Algebras of types B and D

For the simple Lie algebra $\frak{so}_m$, we study the commutant vertex operator algebra of $L_{\hat{\frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $L_{\hat{\frak{so}}_{m}}(1,0)^{\otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{\hat{\frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.Comment: A mistake for the case n=3 is correcte

Classification of holomorphic framed vertex operator algebras of central charge 24

This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.Comment: 26 page

Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras

In this article, we develop a general technique for proving the uniqueness of holomorphic vertex operator algebras based on the orbifold construction and its "reverse" process. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge $24$ is uniquely determined by its weight one Lie algebra if the Lie algebra has the type $E_{6,3}G_{2,1}^3$, $A_{2,3}^6$ or $A_{5,3}D_{4,3}A_{1,1}^3$.Comment: 27 pages. arXiv admin note: text overlap with arXiv:1501.0509

A holomorphic vertex operator algebra of central charge 2424 whose weight one Lie algebra has type A6,7A_{6,7}

In this article, we describe a construction of a holomorphic vertex operator algebras of central charge $24$ whose weight one Lie algebra has type $A_{6,7}$.Comment: 11 page